The Hardy-Littlewood maximal function inequality can be proved using the Vitali covering lemma (infinte version) with constant k=5. Actually it can be shown that the constant in the lemma just needs to be k>3. But I cannot find a counter example why 3 is not enough for the infinite case (it is enough in the finite version of the lemma). Has anyone seen such a counter example?
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$$ Z=\{ B (x,r) : x, r \in R,|x| < \frac {1}{2}, |x| < r <(|x| + 1)\frac {1}{3} \}$$
Then every ball in $Z$ contains $0$ and hence every disjoint subcollection of $ Z$ consists of just one ball. But for any ball $B(x, r)$ , the expanded ball $3B(x, r)$ does not cover $Z = (-1,1) $